It cannot be both if we use original definitions.

No true Scotsman.

In antiquity - which is what the Founders, who (unlike the author of the article apparently) were well-educated, had in mind - democracy meant almost exclusively direct democracy.

First, it doesn't matter what the founders thought democracy and republicanism were, cause we ain't talking about arguments between the founders, we're talking about arguments between people today using contemporary definitions.

Second, you provide no source for this claim; I provided a source in the article, so you should either provide one of your own or back off.

Republic on the other hand is a mixed form of government, with elements of monarchy, aristocracy and democracy.

Is this a joke? The irony here is absurd. First, you say republicanism has elements of monarchy, when the definition of a republic is that it ain't a monarchy, then you say it has elements of democracy, while saying republics can't be democracies.

P.S. actually read the article.